Biomedical Engineering Reference
In-Depth Information
the application, it may be desirable to shift the time-dependent term. For
example, a model equivalent to model (10) is:
E[dN
i
(t)jZ
i
] = d
0
(t) expf
0
Z
i
+ Z
i
(tt
0
)g;
(11)
where t
0
would be chosen to be some readily intuited time; e.g., mean,
median, mid-point of follow-up distribution. Due to the non-proportionality,
the rate ratio (Z
i
= 1 vs. Z
i
= 0) varies with t in model (10) and equals
expf
0
gat t = t
0
in model (11).
4. Simulation Study
To assess the performance of the non-proportional rates model in nite
samples, we conducted a simulation study. A wide range of scenarios were
examined. The number of subjects was set to n = 30, 50, 100 and 200.
Censoring times, C
i
, were generated from a Uniform(0,2.5) distribution.
The simulated non-proportional rates model was given by:
d
0
(t) = Q
i
d
0
expf
0
Z
i
+
0
Z
i
(t1:25)g;
(12)
with d
0
=0.5,
0
= log(2),
0
= 0:5, and Q
i
followed a Gamma distribution
with mean 1 and variance,
2
. The covariate was set to Z
i
= mod(i; 2),
where mod is the remainder operator.
The frailty variate, Q
i
, was included in order to accommodate positive
intra-subject event time correlations. Frailty variances employed included
2
= 0, 0.5, 1.0, and 2.0. For
2
=0, within-subject event times are indepen-
dent. Setting
2
to 0.5 and 1.0 results in positive correlation among event
times for each subject, with
2
=2.0 resulting in extremely strong event time
correlations. In the analysis of recurrent events, at least among human
subjects in biomedical studies, it would be rare to observe
2
=0 or
2
>2.
For each subject, N
i
=25 events were generated from a non-
homogeneous Poisson process (Chiang
3
), with the j'th event time generated
as:
T
i;j
= T
i;j1
1
0
1
0
log(U
i
)
Q
i
d
0
expfZ
i
(
0
1:25
0
)g
log
;
(13)
for j = 1; : : : ; 25, where the U
ij
are Uniform(0,1) variates and T
i;0
0.
Events with (T
i;j
> C
i
) were treated as unobserved.
Due to the non-proportionality (
0
6= 0), the eect of Z
i
on the event
process is not constant over time. The shift in the Z
i
t term allows that
0
represents the log rate ratio at t = 1:25, the mid-point of the observation
period or, equivalently, the mean observation time, since E[C
i
] = 1:25.
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